针对任意边界条件下中心开口矩形板的自由振动特性研究问题，引入改进傅立叶级数方法，用改进傅立叶级数形式表示开口矩形板的位移容许函数，该级数形式具有收敛性好、精度高等特点，采用沿边界均匀分布的线性弹簧模拟任意边界条件，并结合位移连续条件和Rayleigh-Ritz能量泛函变分法，对未知傅里叶展开系数求极值将问题转化为求解一个标准特征值方程问题，通过求解方程可得到中心开口矩形板的固有频率及其对应振型；对不同边界组合不需重新推导公式，只需改变模拟弹簧刚度值即可，提高了效率，最后通过数值算例与有限元方法的计算结果进行对比分析以验证文中方法的有效性和精确性。
Abstract
This paper based on the improved Fourier series method to establish a free vibration analysis model and calculate the natural frequency of rectangular plate with a central opening. When considering the opening, only a quarter of the plate is researched by using the symmetry of rectangular plate with central opening，which is divided into three regions. The admissible function displacement of each region is expressed by the improved Fourier series. By using the linear spring, which is uniform distribution along the border, simulate arbitrary boundary conditions. And through the continuous conditions of displacements determine the relationship of each regional plate. According to the Rayleigh-Ritz energy functional and variational method, we can get the overall energy functional. We can get the generalized eigenvalue matrix equation by studying the extremum of the unknown improved Fourier series expansion coefficients. Solving the equation can obtain the natural frequencies and the corresponding vibration modes of rectangular plate with central opening. Finally, according to the calculation of numerical examples, comparing the calculated results with the finite element method to verify the accuracy and effectiveness of the method in this paper.
关键词
开口矩形板 /
改进傅立叶级数 /
能量法 /
任意边界
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Key words
rectangular plate with opening /
improved Fourier series method /
energy variational method /
arbitrary boundary conditions
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参考文献
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